Class numbers of quadratic extensions of algebraic number fields
type: Item
instance of: scholarly article
title: Class numbers of quadratic extensions of algebraic number fields
zbMATH Open document ID: 0608.12009
author: Jin Nakagawa
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1986-01-01T00:00:00Z
review text: \textit{Y. Yamamoto} [Osaka J. Math. 7, 57-76 (1970; Zbl 0222.12003)] has constructed infinitely many imaginary quadratic fields which have a subgroup of their class group isomorphic to \({\mathbb{Z}}/n\times {\mathbb{Z}}/n\). \textit{H. Naito} [J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 205-211 (1985; Zbl 0582.12004)] has generalized this result to totally imaginary extensions of a given totally real field. In this article the author obtains three results which give infinitely many extensions K of a given field F such that K has a fixed group H as a subgroup of its class group. In the first result K/F is a quadratic extension and H is isomorphic to \({\mathbb{Z}}/n\). For the second result \([K:F]=p\) is an odd prime, F is a quadratic field and H is isomorphic to \({\mathbb{Z}}/n\). In the final result K/F is again quadratic, but H is isomorphic to \({\mathbb{Z}}/3\times {\mathbb{Z}}/3\). In all three results \(N_{K/F}(H)=1\) and some restrictions are placed on the infinite primes of K and F. The proofs employ the diophantine equation which was used in Yamamoto's proof.
reviewed by: Charles J. Parry
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 3983311
zbMATH Keywords: prescribed subgroups of class group
zbMATH Keywords: quadratic extension
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228491
OpenAlex ID: W1981459381
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DOI: 10.2748/TMJ/1178228491
description: scientific article; zbMATH DE number 3983311